Differentiation and the limitations of groupwork

It's important in any profession to stay humble, but teaching has a way of reminding you of this in particularly in-your-face ways, I believe. This semester has really brought home this issue for me in the challenges presented by my upper school Math 3 class. The issues have been around productive groupwork, an area in which I have felt particularly strong and well-trained, so it was perhaps an especially humbling experience to see all of my strategies and approaches come crumbling down and leave me turning to my Twitter network and colleagues to find new ways of helping students work together and feel confident in their progress. I wanted to share and summarize here some of the issues I've worked through that might perhaps be helpful to others.

First, some background.

I have been incorporating the essential elements of a Thinking Classroom in all of my Math classes for the past few years, but most notably in my high school class, where the focus on content and pressures to teach to the test are greater. This year, just like last, I had students read and reflect on Thinking Classrooms and we discussed why most of our time together is spent working on problems in random groups, sharing out ideas and conclusions, and using these to synthesize and summarize learning from the bottom up rather than top down via teacher-led instruction. Students initially seemed bought in and supportive of this type of classroom environment. We set up class norms and discussed the use of group roles, how to step up/step back in group environments, and how to be a skeptical peer and give respectful pushback on ideas.

Several weeks into the semester, however, I started noticing a troubling pattern - some students were disengaging from their collaborative work and seemed very hesitant in sharing their thinking within either small groups or the larger class. Then, I started to hear two different complaints from students - some were feeling that their work during class was unproductive because their groups moved too fast and they were feeling increasingly anxious and uncertain about their mathematical understanding and abilities. They were feeling unprepared to do problems independently on homework assignments or assessments and wanted more teacher guidance and structure, as well as more opportunities to go at their own pace and understand ideas more fully. Other students raised the opposite issue - they felt that the pace of the class was too slow, that they were doing too many problems that they already knew how to do or could figure out quickly and wanted more challenging and deeper problems, both during class and on homework assignments.

When group work goes wrong:

In reflecting on these issues and why they were coming up this year, I realized that we had actually made quite a large change to the Math program without making any changes to our curriculum or pedagogy. This was the first year that we had decided to mix all grades taking a particular Math course - Math had been the only discipline at the Upper School in which students were separated out by grade level. In the past, 9th and 10th graders taking Math 3 (students who had accelerated the normal sequence) were in different sections from 11th graders taking the class (students who were on grade level in terms of their progress through the sequence). This year really was different in terms of prior math experiences, expectations, and desire for challenge/acceleration for students in the same class and the normal groupwork structures were not sufficient to bring together students with such varying backgrounds and approaches.

My next step was to look for feedback from colleagues as well as the Twitter math teacher community. Some suggestions that I implemented that seemed to make a difference:

• Taking a break from random groups to help students regain their trust that the class would meet their needs; doing some work in pairs designed to foster productive collaboration; allowing students choice as to who to work with while also asking them to work with different students at times; being explicit when the goal of a task was to build collaborative skills
• Structuring activities so there was time at the start for individual exploration before asking students to share their thinking with others thus giving more processing time for students who worked more slowly; circulating and helping some students get started; building more optional challenge into tasks for students who worked very quickly or who had already had prior experience with a topic; creating tasks that could be approached with a greater variety of methods and building more writing into tasks so that different ways of thinking mathematically could be valued
• Meeting students where they were to regain trust and buy-in; this included at times splitting the class into two groups (students chose which group to join) - a more free-form exploratory group with more open and challenging problems and a more structured group where students would get some problems to activate prior knowledge and smaller, more concrete problems that would build over time to greater generalization and abstraction and more teacher guidance and reassurance that they were on the right track
• Noticing struggling students' successes and highlighting them publicly; selecting which students would share their thinking to make sure that different voices could be heard over time
• Make sure to leave time for synthesis and practice problems (at different levels) during class - this helped address student concerns that they were leaving class with lots of questions and feeling unsettled about the concepts they had explored that day
• Giving students more feedback during class about their understanding of a topic rather than relying more heavily on groupwork and self-assessment for students to know how they were doing and what might be helpful next steps
• Providing more problems at different levels and helping students navigate which problems might be more helpful for them to do during/after a particular lesson - here is an example of a tiered homework problem set.
• Providing more textbook resources - explicitly linking textbook sections to problem sets for students who wanted more references and examples

We are considering sorting students for next semester by grade level to decrease the heterogeneity of classes - while these strategies have alleviated the issues significantly, it does seem that productive collaboration and exploration is challenged when students in a class are so spread out. Despite these strategies, for example, it usually doesn't make sense for students who have worked quickly and deeply and have figured out challenging extensions to share their ideas with the whole class, most of whom have not even tried these problems. As a result, the class sometimes lacks cohesion and feeling like a true community. Additionally, the amount of work required to run a class with this much differentiation is really, really high. I'm essentially designing at least two different classes, creating both lesson plans and homework assignments that can reach the full spread of student interest and background, and giving individual and frequent feedback to students or small groups of students into which the class has fragmented. This is not really tenable for the whole year given my other preps and teaching responsibilities. However, breaking up students by grade level seems to run counter to our values of equitable access to challenging mathematics for all students and means that Math classes are essentially different from all other classes at the Upper School.

I would welcome any feedback or suggestions that others have around this issue - what strategies have worked for you in working with very heterogeneous groups? 

Connecting Math and CS with probability game simulations

One of my goals for this school year was to build out a few interesting and relevant projects into the 7th grade curriculum, which seemed a bit dry and skill-focused. One area that seemed to beg for an application project was the first unit on Data and Probability. Since one of my other goals was to incorporate more computer science into my classes, it was a no brainer. Developing a cross-over computer science project for this grade level proved to be a bit tricky because students are all over in terms of their experience with programming - we have students who have been coding for years as well as new students who have never coded anything before. I tried to develop a project that would differentiate appropriately and allow students to either explore the CS or the Math parts in greater depth, depending on their interest in and experience with programming.

Here is the project description. You'll notice that I created three distinct strands with different goals and let students select the one that was most appropriate and interesting for them. I was also lucky that the computer science teacher was able to come to my classes for some of the time that students worked on this project. Having many intermediate checkpoints for students to submit pieces of the project was very helpful here in ensuring that I could identify those who were behind or struggling and work with them during class.

Things that I would still like to build out:

• A more robust peer editing process -- I'd like students to be able to present their optimal winning strategy to peers and get critical feedback on how convincing their reasoning is that they would be able to incorporate into their final draft
• A revised rubric to make it more concise
• Move some pieces of this project out to computer science class - this definitely took up quite a bit of time, especially because I felt that most or all of the coding work should happen during class where students would have support
• A clearer division between group and individual aspects - this is always a challenge for me when designing group projects in terms of maximizing student learning and individual accountability. Students seemed to work well together during class, but this isn't an explicit part of the project currently. 
• Some sort of final presentation - for projects like this, I think that having the final product on display or presented to others creates a much more authentic need for clarity and functionality. I haven't figured out a good way to do that for this project. Should students do a gallery walk of projects within the class? Can this be presented or shared with students in other classes somehow? What about with parents?
• Other connections - is this something that can connect to students' work from previous years so that it feels less like a stand-alone project and more like a continuation of ongoing work and thinking? Are there other aspects of this project that can connect to other disciplines, like writing? Can we build on this in future years of either computer science or math curriculum?

Feedback and communicating with families

A goal I wanted to work on this school year is more systematic feedback on mathematical practices as well as better communication with families about what students were working on and their progress. I also wanted to do it in a way that didn't emphasize grading and evaluation and kept the student at the center of setting goals, reflecting on progress, and owning the process.

This blog post had a great suggestion for using Google forms to have students reflect each week and have those reflections emailed to parents. The prompts asked students to describe what they learned that week and how they feel about the class. To be honest, the directions for setting up the emailing were a bit too complicated for me and involved using Add-Ons that our tech administrator wasn't too jazzed about, so I did it in a way that seemed more simple and worked well for me. I'll summarize the deets below, but wanted to first say that I've done this twice now (students are reflecting every other week) and have gotten very positive responses from parents. It takes a lot less time than emailing individual parents, and I think it makes a big difference for parents to hear about progress in their children's own voice.

I changed the questions to be a bit more focused on goal-setting and learning. The questions I'm asking are:

1. What have you learned in the last two weeks? Be as specific as you can - feel free to look through your notebook.
2. How do you feel about your learning of this material, both from class work and homework? (3 = I can teach it to someone else; 2 = I understand it pretty well, but have some questions; 1 = I am very confused and/or have a lot of questions)
3. How do you feel about your class engagement and work? Have you been engaged and focused? Have you worked productively with a variety of classmates? Have you been a respectful skeptical peer and asked for feedback on your thinking?
4. How do you feel about your homework effort? Did you allocate time well during the week? Pick problems at a good challenge level? Stick with hard problems? Try different things? Ask questions? Make corrections during class?
5. What was your goal/next steps the last time you reflected? Did you make progress towards this goal? Why or why not?
6. What are your next steps? What should you keep doing during class and at home? What should you do differently? Do you need to follow up with your teacher?

To clarify, students have a lot of choice in their homework each week - they have an hour to spend on a problem set that has questions at different levels of challenge and depth so I find it helpful for them to reflect on their choices and make changes, if needed. They also have a single assignment due at the end of each week so they should be thinking about how to best allocate their time during the week to avoid leaving it for the last minute.

I make a new version of the form every two weeks and the responses feed into a spreadsheet. I also ask for their name so I can sort the responses alphabetically. I add a column at the end where I add any additional notes I want to share with the family. Usually, it's things like, "This is a great goal. It sounds like X is ready to try some harder problems on the homework next week." I have a list of parent emails that I can then paste in as well as two somewhat fancy things that make the whole system work (not that fancy in actuality, but let me get excited here for a sec). The first one is a cell that combines all of the student responses in one place for ease of emailing. 

The code to make that magic happen is: 

CHAR(10) just creates a line break between responses. The & symbol concatenates responses so that they appear next to the question. Otherwise, it just pulls the responses into a single cell. Drag down the formula to have this for all of the students. Then, add another column to the right that will track whether an email has been sent (this is useful if some students are absent and do this later so you end up running the email script multiple times and don't want to resend the emails that already sent).

When you're done, you have a spreadsheet that looks like this:

(your email sent column will initially be blank)

Okay, this is where the fun really begins. Under Tools, select Script Editor. I found a script for emailing from a spreadsheet and amended it to email two addresses. You can use it too. Ta da.

The code in that link is:

// This constant is written in column C for rows for which an email

// has been sent successfully.

var EMAIL_SENT = 'EMAIL_SENT';

/**

 * Sends non-duplicate emails with data from the current spreadsheet.

 */

function sendEmails2() {

  var sheet = SpreadsheetApp.getActiveSheet();

  var startRow = 1; // First row of data to process

  var numRows = 28; // Number of rows to process

  // Fetch the range of cells desired

  var dataRange = sheet.getRange(startRow, 1, numRows, 4);

  // Fetch values for each column in the Range.

  var data = dataRange.getValues();

  for (var i = 0; i < data.length; ++i) {

    var row = data[i];

    var emailAddress1 = row[0]; // First column

    var emailAddress2 = row[1]; // Second column

    var message = row[2]; // Third column

    var emailSent = row[3]; // Fourth column

    if (emailSent != EMAIL_SENT) { // Prevents sending duplicates

      var subject = 'Bi-Weekly Math Update';

      MailApp.sendEmail(emailAddress1, subject, message);

      MailApp.sendEmail(emailAddress2, subject, message);

      sheet.getRange(startRow + i, 4).setValue(EMAIL_SENT);

      // Make sure the cell is updated right away in case the script is interrupted

      SpreadsheetApp.flush();

    }

  }

}

Notice that my script currently starts on the first row and processes 28 rows (I piloted this in two sections only). You might have more students so will need to process a larger number of rows. You do need to make sure you don't go too far and get to an empty row. The script doesn't like it when there's no data in a cell it's calling up. By the way, when parents respond to this script-generated email, their response goes directly to my regular school email address because Google is magical.

How do students have access to all of their reflections, you ask? I went a bit Google spreadsheet happy and added a tab to my master grading spreadsheet that pulls in the reflection responses for each student using the IMPORTRANGE function. Each student then has their own spreadsheet that pulls in just their reflection responses (as well as feedback on content learning goals). There is now a chain of Google sheets happily talking to each other and emailing parents every two weeks. What a world.

Culture of Mathematics

This post is part of the Virtual Conference on Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures -- from problem solvers and theorists, musicians and artists, explorers, alchemists and wrestlers, to "makers of patterns." Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking. 

I am excited to write a post as part of a group of bloggers thinking about the tension between problem solving and theoretical understanding, among other tensions. Moreover, the benefit of procrastinating and getting terribly behind is that I get to read and respond to some of the other blogs written as part of this group. Michael's post, in which he discusses the reasons that he has moved away from problem solving as a classroom focus, was one that really struck me and prompted me to want to respond. I think that he makes some excellent points about wanting to move away from answer getting as an inherently inequitable and exclusionary practice in which some students race ahead while others are left behind. It's a great read, and I highly recommend you pause here and read his post in full. 

The main place where I found myself disagreeing was in the setup, in which problem-solving is positioned diametrically opposed to theory-building, and the two trade off against each other. This, to me, seems like a confusing and artificial construction... both are just questions that we are posing about the world, where perhaps problem-solving takes the form of slightly more specific questions and theory-building is what we call questions that are more general. Joshua Bowman calls out this false dichotomy in his post as well, adding it to the list of polarities like applied vs. theoretical and individual vs. communal and urging for math teachers to value both types of thinking because we just don't know what's going to motivate or interest a particular student and the more variety and ways there are to be hooked into mathematical thinking, the better. 

I would say that as teachers, we can't help but be biased towards ways of thinking that are aligned to how we ourselves think and what we value. When I first started teaching, I was very much tapping into my own personal experiences as a math student - the complete disconnect I had felt from math as an intellectual discipline in high school and why I fell in love with math as an undergraduate, thinking for the first time about real (to me) mathematical questions that sparked my curiosity and wonder and ideas that blew my mind and made me want to learn more. I posed problems to my high school students in the way that I would have wanted them posed to me. There were some kids who came along for the ride, but there were also definitely some who were left behind because I was not speaking their language.

Joshua's conscious choice to provide students with many options and potential hooks is a way to move away from this form of me-centered teaching, which can be such a natural trap. He chooses to be agnostic and let students construct knowledge in the way that works for them. I find it interesting that Michael is perhaps doing the same thing, but in a way that purposefully deemphasizes problem-solving because it is such a dominant paradigm in mathematics so that students are exposed to other ways of doing math. The sentiment behind these teacher decisions definitely resonates for me, and I think should be central in teacher preparation and planning for courses - what values are you emphasizing in your classroom structures, teacher moves, and curriculum? 

I have certainly seen problem-solving play out in the same troubling ways that Michael referenced in his post - primarily when I have attended math team practice and felt the anxiety I often feel in these types of hyper-competitive-speed-based-publicly-exposed environments. But for me, it isn't problem-solving that's the culprit, but the types of problems that have been posed, the environment in which they are done, and their purpose. For example, I attended PCMI last summer - this is a place where math teachers are solving problems together for hours every day. There is a huge amount of variety in mathematical background knowledge, experience with math teaching, and familiarity with the PCMI style. Yet norms are set and problem sets written in such a way that connections, representations, deep and novel ways of thinking and analyzing, and thoughtful questions are what is valued, resulting in a community that while not quite a mathematical utopia, is pretty damn close. Good problems + clear norms + teacher moves to support norms = learning that aligns to the values of the program and access and motivation for many students.

In my own teaching, I have moved towards student-posed questions and projects as something that more closely matches my values in teaching and moves away from my subjective opinion of what is interesting towards my students' perspectives and interests. I value good problem posing as an opportunity to both pique interest, stimulate thinking, and help students better understand what makes for a good problem so they can move on from problems posed by me to problems they pose themselves. It's much less important to me if the questions they ask are specific (problem-solving) or more general (theory-building) - it's in the asking of questions and seeking to understand and construct the world around them that I see the purpose of my teaching.

End of year celebration of knowledge

Dan Meyer started a discussion on Twitter recently about the unnecessary stress that final exams cause for students at the end of the year, questioning how much insight they really give into student learning. It’s been a helpful reminder that while I definitely agree that high-stakes final exams are terrible, I really don’t have a great system yet for wrapping up the year.

We certainly don't want students feeling like this:

But what makes for a good alternative?

It seems challenging to balance the goal of ending the year with celebration and anticipation of more learning, while also gaining information about retention and content synthesis. I want students to end the year on a high note, feeling positive about their progress and provided with the opportunity to dig deeply into a particular topic, but it would also be great to be able to identify topics from the entire year that would benefit from review and work with them to do that.

In some ideal universe where time doesn't exist and Firefly is still on the air, I would be able to do both: a meaty project in which students can shine and review and an assessment of all of the things. However, even given this bounty of time, I'm not sure that a timed, paper and pencil, silent, individual assessment would really promote the most learning and information for me and students.

So I spent a bunch time the last few weeks reading up on various ideas and here is my current compilation.

• A group whiteboard assessment that looks at problem solving and tying together big concepts from the year, something like what @AlexOverwijk does with his classes:
  

  S “Because of the low stress in this testing environment it reassures me that I know stuff, lets me know what I need to focus on tonight before the individual test, and really just increases my confidence.”
  Me”Glad you feel that way.” pic.twitter.com/fPFx1uD7cf

  
  

  — Alex Overwijk (@AlexOverwijk) May 31, 2018

  This would require careful teacher observation to untangle individual understanding and contribution to the group product, but seems like a much closer fit to what students do in class every day and therefore a more accurate picture of their understanding, as well as obviously being less stressful.
• An annotated portfolio of work throughout the term, which would require students to find evidence of learning for previous topics, identify important connections, revise work, and identify topics that need further attention themselves. I really like this option as it puts the student in the driver's seat. However, this would be fairly time-consuming and likely need students to have been tracking their work throughout the semester. It's something I'm strongly considering for next year. If you do this, I'd love more information - directions, rubrics, advice for someone who wants to try it. How do you make this work in large classes?
• An oral final exam in which each student has a one-on-one interview and discusses their process and reasoning for one or two problems, which @JadeMohrWhite proposed:
  

  My last few yrs in the classroom I did oral finals and loved it. I spent time with each individ S and got to have a math convo with them about what they had learned over the year. They explained reasoning & it wasn’t about right or wrong but the process.

  — Jade White (@JadeMohrWhite) May 22, 2018

  
  This seems great for digging deep into mathematical practices and student thinking, but would only give limited content knowledge information due to time constraints. Building in class time for every student to have a 20 minute interview or so also seems a bit daunting in the end-of-year crunch, but could potentially complement a final project or portfolio assignment, during which students are working relatively independently.
• Final individual project and group presentation. This is the model I'm trying this year in one of my classes. Students selected a topic of personal interest to them that is related to the content in the course and did research and Math work related to this topic. They were then placed into groups based on some possible common threads between projects and created a presentation that highlighted their individual work AND the connections between them, as well as how what they learned related to their Math course this year. Detailed directions are here.
  
  I like how positive and forward-looking the projects have been this year - it does feel like a celebration and memorable opportunity for students to shine. However, because projects are typically looking at a single topic in a great deal of depth, this way of ending the year misses out on the whole cumulative, wrapping everything up feeling that I like to have. 
• Bring back the final exam, but have it be extremely low stakes by focusing on retention, connections, and structured so that it can only help a student's grade, not hurt it. This is how I've done final exams before - as a final opportunity for a student to show understanding of a topic from a previous unit and a place to look at cumulative retention and synthesis. It's efficient and serves that purpose well, but isn't the kind of experience I want students to take away with them as their last memory of my class, so if I brought it back, I would definitely want to pair it with one of the above ideas.
  
• Edited to add:
  
  Take-home final exams, as described by @benjamin_leis below, seem like another way to get more comprehensive information about content knowledge in a less-stressful setting. I like the idea of removing time pressure from the equation and letting students assess in a more comfortable and familiar setting where they can take breaks and dig deeper into problems. Again, because this more closely replicates the ways that students do math in my class during the year, it should be a better assessment of what they know. I also think questions on a take-home final should be more interesting and less routine than what I would ask on an in-class timed assessment. 

I would love to know of other ideas people have for alternatives to high-stakes final exams or any feedback on these still-cooking ones. Share them in the comments or send out a tweet.